A group representation tries to fit the multiplication in the group into a matrix:
g -> D(g)

is a homomorphism with D(g) a order(G)xorder(G) matrix

g1g2->D(g1)D(g2)=D(g1g2)

Group multiplication -> matrix multiplication

D(e)= Identity.

D(g^)=(D(g))^

if D is an isomorphism => representation is faithful. => g1!=g2 => D(g1)!=D(g2)

unfaithful representation => non-trivial kernel, K.
Example:

D2:

D(e)=

| 1 0 |

| 0 1 |

D(a)=

| -1 0 |

| 0 1 |

D(b) =

| 1 0 |

| 0 -1 |

D(ab) =

| -1 0 |

| 0 -1 |

Rotation group of a circle

R(o) =

| cos(o) -sin(o) |

| sin(o) cos(o) |

R(o1)R(o2)=R(o1+o2)

source

jl@crush.caltech.edu index

regular_representation

real_representation

kronecker_product

euclidean_group

direct_product

lorentz_group

irreducible

pseudoreal

equivalent

direct_sum

reducible

character

unitary

tensor

SU

SO